Stochastic Komatu-Loewner evolutions and SLEs

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Let D=H ⱼ₌₁ⁿ Cⱼ be a standard slit domain, where H is the upper half plane and Cⱼ,1≤ j≤ N, are mutually disjoint horizontal line segments in H. A stochastic Komatu-Loewner evolution denoted by SKLEα,b has been introduced in CF as a family {Fₜ} of random growing hulls with Fₜ⊂ D driven by a diffusion process ξ(t) on ∂ H that is determined by certain continuous homogeneous functions α and b defined on the space S of all labelled standard slit domains. We aim at identifying the distribution of a suitably reparametrized SKLEα,b with that of the Loewner evolution on H driven by the path of a certain continuous semimartingale and thereby relating the former to the distribution of SLEα₂ when α is a constant. We then prove that, when α is a constant, SKLEα,b up to some random hitting time and modulo a time change has the same distribution as SLEα₂ under a suitable Girsanov transformation. We further show that a reparametrized SKLE√6,-b BMD has the same distribution as SLE₆, where b BMD is the BMD-domain constant indicating the discrepancy of D from H relative to Brownian motion with darning (BMD in abbreviation). A key ingredient of the proof is a hitting time analysis for the absorbing Brownian motion on H. We also revisit and examine the locality property of SLE₆ in several canonical domains. Finally K-L equations and SKLEs for other canonical multiply connected planar domains than the standard slit one are recalled and examined.